Formulas TFY4345

Validity of formulas and meaning of symbols are assumed to be known.

• Hamilton’s equations:

q̇_{i} = \frac{∂H}{∂p_{i}} , ṗ_{i} = −\frac{∂H}{∂q_{i}}

• Lagrange’s equations:

\frac{d}{dt} \frac{∂L}{∂q̇_{i}} − \frac{∂L}{∂q_{i}} = 0

• Poisson brackets:

[f, g] = \frac{∂f}{∂q_{i}} \frac{∂g}{∂p_{i}} − \frac{∂g}{∂q_{i}} \frac{∂f}{∂p_{i}}

[q_{i}, q_{j}] = [p_{i}, p_{j}] = 0 , [q_{i}, p_{j}] = δ_{ij}

• Four vector:

x_{μ} = (r, ict)

• Four potential:

A_{μ} = (A, iϕ/c)

• Electromagnetic field:

E = −∇ϕ − \frac{∂A}{∂t}

B = ∇ × A

• Lorentz transformation (with relative velocity v = vx̂):

L_{22} = L_{33} = 1 , L_{11} = L_{44} = γ , L_{14} = −L_{41} = iβγ β = v/c , γ = 1/\sqrt{1 − v^{2}/c^{2}}

• Lorentz transformation of electromagnetic field (where the inertial system S_{0} moves with ve-

locity vx̂ with respect to S):

E_{x} = E_{0x} , E_{y} = γ(E_{0y} + vB_{0z}) , E_{z} = γ(E_{0z} − vB_{0y})
B_{x} = B_{0x} , B_{y} = γ(B_{0y} − \frac{v}{c^{2}} E_{0z}) , B_{z} = γ(B_{0z} + \frac{v}{c^{2}} E_{0y})

• Trigonometric relations:

cos(a ± b) = cos a cos b ∓ sin a sin b
sin(a ± b) = sin a cos b ± cos a sin b